First, expand to series t(1+t)/(4+t^4) = t²/4 - (t^3)/8 +O(t^4)
Second, integration from t=0 to x leads to I= (x^3)/12 - (x^4)/32 +O(x^5)
Then, I/x^3 = 1/12 + x/32 +O(x²) which limit for x -> 0 is 1/12
The derivative of the numerator is (using the Fundamental Theorem of Calculs) $\displaystyle \frac{x \ln(1 + x)}{x^4 + 4}$ and the derivative of the denominator is 3x^2. Therefore ....